Research on Mining Subsidence Prediction Parameter Inversion Based on Improved Modular Vector Method

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This paper mainly applies to the parameter inversion problem in mining subsidence under the condition of lack of empirical values.

Abstract

In this study conducted in the Shendong mining area, this paper tackles the challenge of estimating mining subsidence parameters in the absence of empirical values. The study employs a tailored pattern recognition method specifically designed for mining subsidence in a specific working face. The goal is to determine a globally approximate optimal solution for these parameters. Subsequently, this study utilizes the approximate optimal solution as an initial exploration value and harnesses the modular vector method to obtain stable, accurate, optimal solutions for the parameters. The results of the study demonstrate the effectiveness of the improved modular vector method. In simulation tests involving the subsidence coefficient, the main influence angle tangent value, the propagation angle of mining influence, and the deviation of the inflection point, the relative errors do not exceed 1.2%, 1.9%, 1.7%, and 7.9%, respectively. Furthermore, when subjected to random errors of less than 20 mm, the relative errors for each parameter remain below 2%. Even in conditions with 200 mm gross error, the relative error for each parameter does not exceed 5.1%, indicating high precision. In an engineering example, the root mean square error of the improved modular vector method’s fitting result is 64.31 mm, constituting a mere 1.79% of the maximum subsidence value. This performance surpasses that of the genetic algorithm (70.47 mm), particle swarm algorithm (72.82 mm), and simulated annealing algorithm (75.45 mm). Notably, the improved modular vector method exhibits superior stability and reduced reliance on the quantity of measured values compared to the three aforementioned algorithms. The inversion analysis of predicted parameters based on the improved modular vector method, coupled with the probability integral method, holds practical significance for enhancing the accuracy of mining subsidence prediction.

1. Introduction

With the rapid development of the economy, the output of coal resources is increasing year by year. Large-scale coal mining has also brought a series of serious social and environmental problems [1], such as land subsidence, building damage, ground fissures, etc. [2,3,4], which have attracted worldwide attention. Therefore, the accurate prediction of surface movement and deformation is of great significance to the exploitation of coal resources. In the field of mining subsidence research, the probability integral method (PIM) is the most commonly used parameter inversion method for mining subsidence prediction in China. Its theoretical basis is the random medium theory. This theoretical framework was first introduced to the study of mining subsidence by the Polish scholar Litwiniszyn in the 1950s [5]. Subsequently, Chinese scholars, such as Baochen Liu and Guohua Liao, further developed this theory into what is now known as the probability integral method [6]. Currently, the probability integral method stands as one of the more mature and most widely utilized techniques in China for mining subsidence prediction.

Over time, the inversion methods employed for estimating the parameters associated with the probability integral method have evolved significantly. This evolution has progressed from linear approximation methods to experimental design approaches and ultimately to optimization and intelligent algorithms. Due to the complex nature of the probability integral method function, which encompasses numerous unknown parameters and does not adhere to linear requirements, conventional linear approximation methods often face convergence challenges, even when appropriate initial values are selected [7,8,9,10]. On the other hand, the experimental design approach tends to necessitate a substantial number of trials due to the multitude of parameters involved, resulting in increased workload and reduced parameter inversion speed [11]. Consequently, many researchers have sought to employ optimization algorithms and intelligent algorithms in the parameter inversion process. For instance, Zha et al. [12] successfully applied a genetic algorithm to the probability integral method for parameter inversion, demonstrating notable advantages in terms of accuracy and reliability. Subsequent investigations explored the use of the modular vector method [13,14,15], particle swarm algorithms [16,17,18,19], simulated annealing algorithms [20,21], and others [22,23,24,25,26,27] for parameter inversion within the probability integral method framework, all yielding highly satisfactory results. In a comparative analysis of parameter inversion outcomes using various intelligent algorithms, Han Mei et al. [28] confirmed that, with an appropriate choice of initial exploration values, the modular vector method excels in accuracy and reliability when contrasted with other algorithms. Nevertheless, it is important to note that different selections of initial exploration values may lead to cases where the fitting error is minimal, yet, the results of parameter inversion exhibit substantial discrepancies.

To mitigate the risk of erroneous initial exploration value selection leading to the confinement of the modular vector method’s inversion results within local extremities, this paper introduces an improved algorithm, which amalgamates the modular vector method with the pattern recognition approach [29,30] in mining subsidence analysis. The algorithm leverages geological data from the mining area to determine the initial exploration values of the modular vector method. This study first verifies the accuracy and robustness of the improved modular vector method in the inversion of anticipated parameters within the probability integral method through simulation experiments conducted under geological conditions characteristic of shallow-buried coal seams [31]. Subsequently, an illustrative analysis is conducted, focusing on a specific working face within the Shendong mining area. The analysis serves to underscore the distinctive advantages of the improved modular vector method compared to genetic algorithms, particle swarm algorithms, and simulated annealing algorithms. These advantages manifest in three key dimensions: the precision of inversion results, stability, and the degree of reliance on the quantity of measured data.

In conclusion, the presented algorithm offers a robust solution to mitigate the local extrema issues encountered in the modular vector method resulting from inadequate initial exploration value selection. The research outcomes, demonstrated through simulation experiments and engineering case studies, underscore the effectiveness of the improved modular vector method in addressing the challenges of parameter inversion in the probability integral method when empirical values are lacking.

2. Theory of Parameter Inversion

2.1. Prediction Principles of the Probability Integral Method

The probability integral method, initially developed by prominent scholars, such as Baochen Liu and Guohua Liao, is rooted in the theory of random medium. Over time, this method has seen substantial refinement and enhancement through continuous scholarly contributions, solidifying its position as the predominant approach for predicting mining subsidence.

The subsidence value at any point within a surface subsidence movement basin induced by mining subsidence can be represented as follows [32,33]:

$$W(x,y)=\frac{1}{W_0}\times W^0(x)\times W^0(y)$$

(1)

$$W_0=m\times q\times \cos \alpha$$

(2)

$$\begin{array}{c}{W}^0(x)=\frac{W_0}{2}\times [erf(\frac{\sqrt{\pi }\times tg\beta }{H}\times x)+1]-\frac{W_0}{2}\times [erf(\frac{\sqrt{\pi }\times tg\beta }{H}\times \\ (x-(D_3-S_1-S_2)))+1]\end{array}$$

(3)

$$\begin{array}{c}{W}^0(y)=\frac{W_0}{2}\times [erf(\frac{\sqrt{\pi }\times tg{\beta }_1}{H_1}\times y)+1]-\frac{W_0}{2}\times [erf(\frac{\sqrt{\pi }\times tg{\beta }_2}{H_2}\times \\ (y-(D_1-S_3-S_4)\times \frac{\sin (\theta +\alpha )}{\sin \theta }))+1]\end{array}$$

(4)

In the formula provided, $W_0$ represents the maximum subsidence value of the surface when both the strike and the dip reach full mining. $W^0(x)$ signifies the subsidence values on the strike primary section when the dip attains full mining. $W^0(y)$ denotes the subsidence values on the dip primary section when the strike direction reaches full mining. $m$ stands for the thickness of coal seam mining. $q$ corresponds to the subsidence coefficient. $\alpha$ represents the dip angle of the coal seam. $\theta$ represents the propagation angle of mining influence. $H$, $H_1$, and $H_2$ denote the average mining depth in the strike direction, the mining depth in the dip downhill direction, and the mining depth in the dip uphill direction, respectively. $D_1$ signifies the dip length of the working face. $D_3$ represents the length of the working face strike. $S_1$, $S_2$, $S_3$, and $S_4$ represent the deviation of the inflection point of the left and right boundaries and the deviation of the inflection point in the uphill and downhill directions, respectively. $tg\beta$, $tg{\beta }_1$, and $tg{\beta }_2$ denote the main influence angle tangent value on the strike line, the main influence angle tangent value in the dip uphill direction, and the dip downhill direction, respectively. The spatial mining configuration is illustrated in Figure 1.

2.2. Principles and Optimization Steps of the Improved Modular Vector Method

The modular vector method, also known as the step acceleration method, is a technique utilized to expedite the optimization of the valley line (ridge line) direction. This method presents notable advantages, particularly in addressing unconstrained extreme value problems [34].

In the context of parameter inversion within the probability integral method using the modular vector method, precautions are taken to prevent the inversion results from converging toward local extrema due to inappropriate initial exploration value selection. To address this concern, a pattern recognition approach based on the fuzzy pattern recognition theory is employed. This approach utilizes mining subsidence pattern recognition techniques and adheres to a specified classification of similar phenomenon groups in mining subsidence and fundamental rock movement parameters. The preliminary determination of mining subsidence rock movement parameters for the mining area is based on the principle of selecting the closest matching parameters.

The procedural steps for mining subsidence pattern recognition and parameter determination are as follows:

(1)

Calculate the comprehensive deformation modulus of the rock mass using drilling data obtained from the specific mining area.

(2)

Compute the similarity criterion for the mining area in question.

(3)

Determine the average similarity criterion across all types of mining subsidence.

(4)

Calculate the proximity degree between the mining area in question and each type of mining subsidence. Following the principle of maximum proximity degree, identify the category to which the mining area belongs. Adopt the mining subsidence rock movement parameters corresponding to the determined mining subsidence category for the mining area under investigation.

Following the prediction principle of the probability integral method, it is evident that the relationship between the subsidence value at any designated point $(x,y)$ on the surface and the predictive parameters of the probability integral method can be expressed as follows:

$$f(x,y)=W(x,y;B)$$

(5)

In the formula, $B$ represents the predicted parameters of the probability integral method, which are the values subject to inversion. $B={(q,\theta ,tg\beta ,S_1,S_2,S_3,S_4)}^T$.

Assume that $F(x,y)$ represents the measured values at a specific point $(x,y)$ on the surface. Following the principle of least squares, the criterion for parameter inversion relies on minimizing the sum of squared errors between the predicted values and the measured values, which can be expressed as follows:

$${\displaystyle \sum _{i=1}^n{v}_i{}^2}=\min$$

(6)

$$v_i=F(x,y)-f(x,y)$$

(7)

In accordance with Equation (6), the general procedural steps for constructing the modular vector method for parameter inversion within the probability integral method are as follows:

(1) Formulate the objective function as follows:

$$\epsilon (B)={\displaystyle \sum _{i=1}^n{v}_i{}^2}$$

(8)

In the equation, $\epsilon (B)$ represents the predicted error function.

(2) Determine the initial parameter values: $B_1={(q,\theta ,tg\beta ,S_1,S_2,S_3,S_4)}^T$.

(3) Specify the exploration step size and termination step size for each parameter.

(4) Compute the error function value $\epsilon (B_1)$ at the initial reference point $B_1$, then adjust the modulus vector of the first parameter in $B_1$ to obtain the new modulus vector point $B_{11}$. The calculation of $B_{11}$ is in accordance with Equation (9).

$$B_{11}=\left\{\begin{array}{l}\begin{array}{cc}{B}_1+{\Delta }_{11}& ,\epsilon (B_1+{\Delta }_{11})<\epsilon (B_1)\end{array}\\ \begin{array}{cc}{B}_1-{\Delta }_{11}& ,\epsilon (B_1-{\Delta }_{1i})<\epsilon (B_1)<\epsilon (B_1+{\Delta }_{11})\end{array}\\ \begin{array}{cc}{B}_1& ,\epsilon (B_1)<\min \{\epsilon (B_1-{\Delta }_{1i}),\epsilon (B_1+{\Delta }_{11})\}\end{array}\end{array}\right.$$

(9)

In the equation, ${\Delta }_{11}$ signifies the current exploration step size for $B_{11}$.

(5) $B_{11}$ is employed as the interim modulus vector point, with the remaining parameters of $B_1$ being sequentially adjusted. After each adjustment, the resulting new modulus vector point is treated as the interim modulus vector point for the subsequent adjustment. The calculation of this new point, denoted as $B_{1i}$, is determined in accordance with Equation (10).

$$B_{1i}=\left\{\begin{array}{l}\begin{array}{cc}{B}_{1(i-1)}+{\Delta }_{1i}& ,\epsilon (B_{1(i-1)}+{\Delta }_{1i})<\epsilon (B_{1(i-1)})\end{array}\\ \begin{array}{cc}{B}_{1(i-1)}-{\Delta }_{1i}& ,\epsilon (B_{1(i-1)}-{\Delta }_{1i})<\epsilon (B_{1(i-1)})<\epsilon (B_{1(i-1)}+{\Delta }_{1i})\end{array}\\ \begin{array}{cc}{B}_{1(i-1)}& ,\epsilon (B_{1(i-1)})<\min \{\epsilon (B_{1(i-1)}-{\Delta }_{1i}),\epsilon (B_{1(i-1)}+{\Delta }_{1i})\}\end{array}\end{array}\right.$$

(10)

In the formulae, ${\Delta }_{1i}$ represents the current exploration step size of $B_{1i}$, and i ranges from 2 to 7.

(6) Once all parameters have been adjusted using the modulus vector movement, the temporary modulus vector point $B_{17}$ is achieved. By setting $B_2$ equal to $B_{17}$, the first modulus vector $B_{20}$ is constructed from the new base point $B_2$ and the initial base point $B_1$, with the calculation of $B_{20}$ following the procedure outlined in Equation (11).

$$B_{20}=B_1+2(B_2-B_1)$$

(11)

(7) An analogous exploration within the vicinity of $B_{20}$ results in the acquisition of temporary modulus vector points $B_{21}$, $B_{22}$, … $B_{27}$. In this context, $B_3$ is set equal to $B_{27}$, and the second modulus vector, denoted as $B_{30}$, is determined based on $B_3$ and $B_2$. The calculation of $B_{30}$ is elaborated in Equation (12).

$$B_{30}=B_2+2(B_3-B_2)$$

(12)

(8) Iteratively repeat the aforementioned process until the jth modulus vector satisfies $\epsilon (B_{j0})<\epsilon (B_j)$. If necessary, reduce the step size and continue exploration until the termination step size condition is met. Upon satisfying the termination criteria, conclude the iteration process and output the optimal solution.

3. Accuracy and Robustness Simulation of the Improved Modular Vector Method

3.1. Simulation Scheme

Taking into consideration the geological and mining characteristics of a shallow-buried near-horizontal coal seam [35,36], the simulation involves a working face with a dip length of 100 m, a strike length of 200 m, a coal seam dip angle of 0 degrees, an average mining depth of 100 m, a coal seam thickness of 4 m, and the implementation of the all-roof caving mining method. The initial exploration values for the modular vector method are determined through the application of the mining subsidence pattern recognition method.

Table 1. Empirical values of prediction parameters and the condition values of the improved modular vector method.

Category	$\mathit{q}$	$\mathit{t}\mathit{g}\mathit{\beta }$	$\mathit{\theta }$(°)	${\mathit{S}}_1$/m	${\mathit{S}}_2$/m	${\mathit{S}}_3$/m	${\mathit{S}}_4$/m
Empirical value	0.90	2.14	78	14	14	14	14
Initial exploration value	0.85	2.00	75	15.5	15.5	15.5	15.5
Exploration step size	0.10	0.10	1.00	1.00	1.00	1.00	1.00
Termination step size	0.01	0.01	0.01	0.10	0.10	0.10	0.10

presents the empirical values for the predicted parameters of the probability integral method and the associated conditions for the improved modular vector method. To facilitate the simulation, a total of twenty-one observation points are strategically positioned along the dip direction above the working face. The position relationship between the simulated working face and the monitoring observation points is depicted in Figure 2.

3.2. Accuracy Analysis of Parameter Inversion

The expected subsidence values, determined based on empirical values, serve as the measured data for the inversion of the predicted parameters within the probability integral method. Alongside information pertaining to the working face, the initial exploration values, as determined by the pattern recognition method in mining subsidence, are utilized to invert the parameters of the improved modular vector method.

Upon examination of the parameter inversion results (

Table 2. Parameter inversion results.

Category	$\mathit{q}$	$\mathit{t}\mathit{g}\mathit{\beta }$	$\mathit{\theta }$(°)	${\mathit{S}}_1$/m	${\mathit{S}}_2$/m	${\mathit{S}}_3$/m	${\mathit{S}}_4$/m
Empirical value	0.90	2.14	78.0	14.00	14.00	14.00	14.00
Inversion value	0.91	2.18	76.7	13.75	13.75	13.25	12.90
Relative error/%	1.11	1.87	1.67	1.79	1.79	5.36	7.86

) and the root mean square error of the fitting (Figure 3), it is evident that the root mean square error for the predicted parameters of the improved modular vector method inversion within the probability integral method is merely 1.19 mm. Additionally, the relative errors associated with the subsidence coefficient, the main influence angle tangent value, the propagation angle of mining influence, and the deviation of the inflection point do not exceed 1.2%, 1.9%, 1.7%, and 7.9%, respectively. These findings affirm the accuracy of the improved modular vector method in parameter inversion for the probability integral method.

3.3. Analysis of Anti-Random-Error Interference Ability

Table 2 provides information regarding the initial exploration values, exploration step size, and termination step size for the improved modular vector method. To assess its resilience against random errors, the expected subsidence values for observation points on the working surface are adjusted based on the original expected results. Artificially introduced mean square errors of ±10 mm and ±20 mm, and the subsidence values with these errors, are employed for parameter inversion.

Upon reviewing the inversion results (Figure 4), it becomes apparent that as the error in the subsidence value increases, the relative errors of the parameters inverted by the improved modular vector method exhibit a general upward trend. When the random error remains below 20 mm, the relative errors for the subsidence coefficient $(q)$, the main influence angle tangent value $(tg\beta )$, the propagation angle of mining influence $(\theta )$, and the sum of inflection point offsets $(S)$ all remain below 2%. During data collection, the observation results are inevitably influenced by various factors. The probability integral method, leveraging the improved modular vector method for parameter inversion, proves to be effective in robustly handling the cumulative impact of these factors within a defined range.

3.4. Analysis of Anti-Gross-Error Interference Ability

The initial exploration values, exploration step size, and termination step size for the improved modular vector method remain consistent with those detailed in Section 3.1. In this analysis, the measured data are intentionally subjected to gross error, with a total of two gross errors introduced [37]. Specifically, two 200 mm gross errors are added at both ends, inflection points, and the maximum sinking points, respectively, for the purpose of parameter inversion using the improved modular vector method.

Upon reviewing the inversion results (Figure 5), it becomes evident that the relative error of the improved modular vector method for inversion results with gross error at the inflection point is notably higher than that observed at both ends and the maximum sinking point. In general, when two 200 mm gross errors are introduced, the relative errors for the subsidence coefficient $(q)$, the main influence angle tangent value $(tg\beta )$, the propagation angle of mining influence $(\theta )$, and the sum of inflection point offsets $(S)$ all remain below 5.1%. This outcome verifies that the improved modular vector method possesses a certain degree of resilience against gross error interference. In practical observation scenarios, it can effectively mitigate the influence of human factors on inversion results.

4. Engineering Example

4.1. Geological and Mining Conditions of the Working Face and Initial Exploration Value Acquisition for Parameter Inversion

The research mining area is located on the Shaanxi side at the junction of Shaanxi Province and the Inner Mongolia Autonomous Region, and it is under the jurisdiction of Daliuta Town, Shenmu City. A working face is characterized by an average mining depth of 145 m, a coal seam thickness of 4.4 m, a coal seam dip angle ranging from 1 to 3 degrees, a 325 m dip mining width, and a 2430 m strike length. The lithology of the overlying strata on the working face is detailed in

Table 3. Lithology of overlying strata on the working face.

Overlying Strata	Layer Thickness/m	Stability Coefficient	Delamination Deformation Modulus/MPa	Layered Mass Density /(g/cm3)
Loose layer	12	0.7	360	1.80
Fine sandstone	8	4.5	2721	2.56
Medium stone	14	5.0	3095	2.59
Silty sandstone	10	7.8	5571	2.62
Fine sandstone	10	4.5	2721	2.56
Silty sandstone	8	7.8	5571	2.62
Fine sandstone	12	4.5	2721	2.56
Medium stone	14	5.0	3095	2.59
Grit sandstone	18	6.0	3900	2.63
Silty sandstone	4	7.8	5571	2.62
$1^{-2}$ upper coal	3	2.0	1100	1.44
Fine sandstone	12	4.5	2721	2.56
Medium stone	12	5.0	3095	2.59
$1^{-2} \mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}$	4	2.0	1100	1.44
Fine sandstone	4	4.5	2721	2.56

. Additionally, an observation line runs along the surface in the dip direction of the working face, featuring 22 measuring points, as depicted in Line C in Figure 6.

Based on the information provided in Table 3, the comprehensive deformation modulus for the rock strata within the mining area is determined to be 3129.30 MPa, with an average mass density of 2.469 g/cm³. The similarity criterion for the mining area, as well as the mining subsidence model of the mining area and the closeness degree to various mining subsidence categories, are calculated.

According to the calculation results of the category closeness degree (

Table 4. The calculation results for the category of ‘Closeness degree’.

Category	I	II-1	II-2	II-3	II-4	II-5	II-6	III
Closeness degree	0.562	0.702	0.783	0.948	0.946	0.859	0.745	0.838

Note: In Table 4, I represents extremely weak rock mass with a medium and small depth ratio; II-1 represents relatively weak rock mass with a medium and small depth ratio; II-2 represents weak rock mass with a small depth ratio; II-3 represents weak to medium-hard rock mass with a medium depth ratio; II-4 represents medium-hard rock mass with a medium depth ratio; II-5 represents medium-hard to hard rock mass with a medium depth ratio; II-6 represents hard rock mass with a medium depth ratio; and III represents medium-hard to hard rock mass with a large depth ratio.

), the mining area exhibits the highest closeness degree with category II-3. Consequently, the initial predicted parameters for mining subsidence in the mining area can be approximated as follows: $q$ = 0.85, $b$ = 0.28, $tg\beta$ = 2.00, and $S/H$ = 0.155. Furthermore, considering the ‘Regulations for Coal Pillar Retaining and Coal Mining in Buildings, Water Bodies, Railways, and Main Roadways’ and the specific mining conditions of the working face, adjustments are made to the deviation of the inflection point and the propagation angle of mining influence. These refinements lead to the final initial exploration values for exploration using the modular vector method, denoted as $B_1$.

$$B_1={(q,\theta ,tg\beta ,S_1,S_2,S_3,S_4)}^T={(0.85,{75}^{°},2.00,20,20,15,15)}^T$$

(13)

4.2. Program Design for Parameter Inversion Algorithm

To assess the accuracy and reliability of the estimated parameters obtained through the improved modular vector method for inversion in the probability integral method, MATLAB R2016b software is used to compile the procedures for the improved modular vector method (IMVM), genetic algorithm (GA), particle swarm algorithm (PSA), and simulated annealing algorithm (SAA). These algorithms are based on the working face information and surface movement measurement data. The program is subsequently verified, and the inversion process is illustrated in Figure 7.

4.3. Accuracy Analysis of the Algorithms

The initial exploration values, exploration step size, and termination step size for the improved modular vector method are presented in

Table 5. Condition values of the improved modular vector method.

Category	$\mathit{q}$	$\mathit{t}\mathit{g}\mathit{\beta }$	$\mathit{\theta }$(°)	${\mathit{S}}_1$/m	${\mathit{S}}_2$/m	${\mathit{S}}_3$/m	${\mathit{S}}_4$/m
Initial exploration value	0.85	2.00	75	20	20	15	15
Exploration step size	0.10	0.10	1.00	1.00	1.00	1.00	1.00
Termination step size	0.01	0.01	0.01	0.10	0.10	0.10	0.10

[38,39]. For the genetic algorithm, the parameter ranges are displayed in

Table 6. Parameter values of the genetic algorithm.

Category	$\mathit{q}$	$\mathit{t}\mathit{g}\mathit{\beta }$	$\mathit{\theta }$(°)	${\mathit{S}}_1$/m	${\mathit{S}}_2$/m	${\mathit{S}}_3$/m	${\mathit{S}}_4$/m
Lower limit of value	0.50	1.00	60	0	0	0	0
Upper limit of value	1.00	3.00	90	40	40	40	40

, with a population size of 100, 200 generations, a cross-over probability of 0.95, and a mutation probability of 0.005 [40,41]. Regarding the particle swarm algorithm, the parameter ranges and maximum flight speed are specified in

Table 7. Parameter values and maximum flight speed of the particle swarm algorithm.

Category	$\mathit{q}$	$\mathit{t}\mathit{g}\mathit{\beta }$	$\mathit{\theta }$(°)	${\mathit{S}}_1$/m	${\mathit{S}}_2$/m	${\mathit{S}}_3$/m	${\mathit{S}}_4$/m
Lower limit of value	0.50	1.00	60	0	0	0	0
Upper limit of value	1.00	3.00	90	40	40	40	40
Max flight speed	0.1	0.1	1	2	2	2	2

, with a population size of 100, a maximum of 200 iterations, an inertia weight of 0.90, a particle position update constraint factor of 1, and weight coefficients of 2 for tracking individual optimal positions and group optimal positions [42,43]. The simulated annealing algorithm is initialized with a temperature of 700, which is consistent with the initial parameters of the improved modular vector method. The coefficient of temperature drop is set to 0.95, and the self-adaptive coefficient is also 0.95 [44,45]. To ensure the reliability of the experimental results, 20 experiments are conducted for each of the mentioned algorithms, and the outcomes with the smallest root mean square error are selected for comparison, as depicted in Figure 8.

Figure 8 reveals that the fitting root mean square error of the improved modular vector method is 64.31 mm, surpassing the fitting results of the genetic algorithm (70.47 mm), particle swarm optimization algorithm (72.82 mm), and simulated annealing algorithm (75.45 mm). Notably, the fitting root mean square error of the improved modular vector method accounts for only 1.79% of the maximum measured subsidence value (3598 mm). This outcome substantiates the accuracy of the fitting results achieved by the improved modular vector method.

The inversion results of the improved modular vector method are detailed in

Table 8. Inversion results of the improved modular vector method.

Category	$\mathit{q}$	$\mathit{t}\mathit{g}\mathit{\beta }$	$\mathit{\theta }$(°)	${\mathit{S}}_1$/m	${\mathit{S}}_2$/m	${\mathit{S}}_3$/m	${\mathit{S}}_4$
Initial exploration value	0.85	2.00	75	20	20	15	15
Inversion value	0.85	2.18	72	17.7	17.7	10.9	11.9
Relative error/%	0.00	9.00	4.00	11.5	11.5	27.3	20.7

, showcasing relative errors in the inversion outcomes $q$, $tg\beta$, and $\theta$, which do not exceed 10%. This evidence supports the alignment of the initial exploration values—as determined by the pattern recognition of mining subsidence—with the inversion results, to a certain extent.

4.4. Stability Analysis of the Algorithm

During the parameter inversion process, the parameter values for the improved modular vector method (IMVM), genetic algorithm (GA), particle swarm algorithm (PSA), and simulated annealing algorithm (SAA) remain consistent with those outlined in Section 4.3. Each algorithm is executed 20 times, and the fluctuations in the inversion results for each parameter across different algorithms—along with the average value of the root mean square error between the measured and predicted values in these 20 parameter inversions—are depicted in Figure 9.

Figure 9 analysis reveals that the parameter inversion results maintain stability when using the same initial exploration values for the modular vector method. Conversely, the parameter inversion values for the other three algorithms exhibit fluctuations across the 20 experiments. This suggests that while the genetic algorithm, particle swarm algorithm, and simulated annealing algorithm can ensure the accuracy of inversion results to some extent, they do not guarantee parameter inversion value stability. The improved modular vector method, with its determinate initial exploration values, ensures parameter inversion value stability. Among the 20 experiments, the improved modular vector method exhibits the lowest average root mean square error between the measured and predicted values at 64.31 mm, demonstrating that the method remains stable while achieving a high level of accuracy. In contrast, the simulated annealing algorithm records the highest average root mean square error at 84.06 mm, signifying the lowest stability.

4.5. Analysis of Algorithm Dependence on the Number of Measured Values

To investigate the dependence of the algorithms on the number of measured values, we employed the last 10 measured points as the data support for parameter inversion. The parameters obtained through inversion were then used to predict the values for the first 12 measured points. We calculated the error between these predicted values and the actual measured values of the first 12 points. This analysis aimed to assess the sensitivity of the four algorithms to variations in the number of measured values. To minimize the impact of algorithmic fluctuations, we conducted 20 experiments for each of the four algorithms.

Figure 10 illustrates the minimum values and the average results from these 20 experiments. Notably, the root mean square error of the improved modular vector method is consistently smaller than that of the genetic algorithm, particle swarm algorithm, and simulated annealing algorithm. This finding underscores that the improved modular vector method exhibits less dependence on the number of measured values than the other three algorithms. Consequently, it possesses a superior capability of withstanding the loss of measuring points. In practical situations with challenging observation conditions, reducing the number of observation points can alleviate workload while still yielding accurate inversion results.

5. Discussion

5.1. Analysis of Simulation Test Results

The simulation experiments conducted under shallow-buried near-horizontal coal seam conditions provided valuable insights into the performance of the improved modular vector method. The results indicate that this method exhibits exceptional accuracy and robustness when estimating the expected parameters of the probability integral method within the simulated test environment. However, it is worth noting that further research is required to investigate the performance of parameter inversion under more complex coal seam conditions.

5.2. Engineering Example Analysis

In the engineering example, we sought to further validate the capabilities of the improved modular vector method. Our findings confirm that this method outperforms genetic algorithms, particle swarm optimization algorithms, and simulated annealing algorithms in three critical aspects: accuracy, stability, and dependence on the quantity of measured data for parameter inversion results. Furthermore, there is a degree of consistency observed between the inversion values and the initially selected exploration values. However, it is important to acknowledge that the absence of reference values for the expected parameters limits our ability to conduct a comprehensive analysis of relative errors in the inversion results.

5.3. Influence of Parameter Selection

The selection of parameters within the modular vector method, particularly the initial values, plays a pivotal role in ensuring the accuracy of inversion results. Improper parameter selection can lead to convergence toward local extrema. To address this challenge, we introduced a pattern recognition method based on the fuzzy pattern recognition theory. This method aids in determining the initial values for the modular vector method. Key to the success of the pattern recognition method is conducting a detailed geological survey of the mining area and acquiring lithological information regarding the overlying strata of the mining coal seam.

5.4. Stability and Reliability of Results

In our analysis, we assessed the accuracy and stability of the parameter inversion results obtained using the improved modular vector method and other intelligent algorithms. To account for the fluctuation observed in the inversion results of genetic algorithms, particle swarm optimization algorithms, and simulated annealing algorithms, we conducted 20 experiments for each of these three algorithms in our analysis. By comparing the optimal and average values of the inversion results, we enhanced the reliability of our experimental findings.

6. Conclusions

In this study, we introduced and evaluated an improved modular vector method for the parameter inversion of the probability integral method in the context of mining subsidence prediction. Our findings can be summarized as follows:

(1)

The improved modular vector method, which incorporates pattern recognition techniques for initial parameter estimation, demonstrated superior accuracy and reliability compared to other intelligent algorithms, such as GA, PSA, and SAA. The root mean square error of the improved method was only 1.79% of the measured maximum subsidence value, validating its accuracy.

(2)

The algorithm exhibited robustness against both random and gross error in the measured data. Even in the presence of significant errors, the relative errors of the inverted parameters remained within acceptable limits, highlighting the method’s ability to mitigate data-related uncertainties.

(3)

The stability analysis revealed that the improved modular vector method maintained consistent parameter inversion results across multiple experiments, in contrast to other algorithms, which exhibited more significant fluctuations. This stability enhances the method’s reliability and practicality in real-world applications.

(4)

The method displayed reduced sensitivity to the number of measured values, making it less dependent on data availability. This characteristic allows for the reduction in observation points in challenging field conditions without sacrificing accuracy, thereby lowering the workload and maintaining result accuracy.

In conclusion, the improved modular vector method represents a promising method for parameter inversion of the probability integral method under the condition of lack of empirical values, offering enhanced accuracy, robustness, and stability. Its reduced reliance on measured data quantity makes it particularly suitable for practical applications in challenging field environments. Further research and validation across diverse geological and mining conditions are recommended to consolidate its efficacy and applicability in a broader context.